axiom

axiom

axiom, in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g., "Two things equal to the same thing are equal to each other"; "If equals are added to equals, the sums are equal") and those related to operations (e.g., the associative law and the commutative law). A postulate, like an axiom, is a statement that is accepted without proof; however, it deals with specific subject matter (e.g., properties of geometrical figures) and thus is not so general as an axiom. It is sometimes said that an axiom or postulate is a "self-evident" statement, but the truth of the statement need not be evident and may in some cases even seem to contradict common sense. Moreover, a statement may be an axiom or postulate in one deductive system and may instead be derived from other statements in another system. A set of axioms on which a system is based is often wished to be independent; i.e., no one of its members can be deduced from any combination of the others. (Historically, the development of non-Euclidean geometry grew out of attempts to prove or disprove the independence of the parallel postulate of Euclid.) The axioms should also be consistent; i.e., it should not be possible to deduce contradictory statements from them. Completeness is another property sometimes mentioned in connection with a set of axioms; if the set is complete, then any true statement within the system described by the axioms may be deduced from them.

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).

Logical axioms are usually statements that are taken to be universally true (e.g. A and B implies A), while non-logical axioms (e.g, ) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in that sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

Outside logic and mathematics, the term "axiom" is used loosely for any established principle of some field.

Etymology

The word "axiom" comes from the Greek word ἀξίωμα (axioma), a verbal noun from the verb ἀξιόειν (axioein), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος (axios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greekphilosophers an axiom was a claim which could be seen to be true without any need for proof.

Historical development

Early Greeks

The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference), was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems, if we are talking about mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid.

The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's posterior analytics is a definitive exposition of the classical view.

An “axiom”, in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that

When an equal amount is taken from equals, an equal amount results.

At the foundation of the various sciences lay certain additional hypotheses which were accepted without proof. Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Indeed, Aristotle warns that the content of a science cannot be successfully communicated, if the learner is in doubt about the truth of the postulates.

The classical approach is well illustrated by Euclid's Elements, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).

;Postulates

# It is possible to draw a straight line from any point to any other point.

# It is possible to produce a finite straight line continuously in a straight line.

# ("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.

;Common notions:

# Things which are equal to the same thing are also equal to one another.

# If equals be added to equals, the wholes are equal.

# If equals be subtracted from equals, the remainders are equal.

# Things which coincide with one another are equal to one another.

# The whole is greater than the part.

Modern development

A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. This abstraction, one might even say formalization, makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts.

Structuralist mathematics goes farther, and develops theories and axioms (e.g. field theory, group theory, topology, vector spaces) without any particular application in mind. The distinction between an “axiom” and a “postulate” disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate we get theories that have meaning in wider contexts, hyperbolic geometry for example. We must simply be prepared to use labels like “line” and “parallel” with greater flexibility. The development of hyperbolic geometry taught mathematicians that postulates should be regarded as purely formal statements, and not as facts based on experience.

When mathematicians employ the axioms of a field, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.

It is not correct to say that the axioms of field theory are “propositions that are regarded as true without proof.” Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.

Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and logic itself can be regarded as a branch of mathematics. Frege, Russell, Poincaré, Hilbert, and Gödel are some of the key figures in this development.

In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.

It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of Euclidean geometry, and the related demonstration of the consistency of those axioms.

The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms (Peano's axioms, for example) to construct a statement whose truth is independent of that set of axioms. As a corollary, Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory.

It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of natural numbers, an infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern Zermelo-Frankel axioms for set theory. The axiom of choice, a key hypothesis of this theory, remains a very controversial assumption. Furthermore, using techniques of forcing (Cohen) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo-Frankel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.

Mathematical logic

In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively)

Logical axioms

These are certain formulas in a formal language that are universally valid, that is, formulas that are satisfied by every assignment of values. Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense.

Examples

Propositional logic

In propositional logic it is common to take as logical axioms all formulae of the following forms, where phi, chi, and psi can be any formulae of the language and where the included primitive connectives are only "neg" for negation of the immediately following proposition and "to" for implication from antecedent to consequent propositions:

phi to (psi to phi)

(phi to (psi to chi)) to ((phi to psi) to (phi to chi))

(lnot phi to lnot psi) to (psi to phi).

Each of these patterns is an axiom schema, a rule for generating an infinite number of axioms. For example, if A, B, and C are propositional variables, then A to (B to A) and (A to lnot B) to (C to (A to lnot B)) are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with modus ponens.

Other axiom schemas involving the same or different sets of primitive connectives can be alternatively constructed.

These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed to include a quantifier in the calculus.

Mathematical logic

This means that, for any FirstOrderLanguagex,, the formula x = x, can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by x = x, (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol =, has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.

Another, more interesting example axiom scheme, is that which provides us with what is known as Universal Instantiation:

Axiom scheme for Universal Instantiation. Given a formula phi, in a first-order language mathfrak{L},, a variable x, and a Termt, that is substitutable for x, in phi,, the formula

forall x phi to phi^x_t

is universally valid.

Where the symbol phi^x_t stands for the formula phi, with the term t, substituted for x,. (See variable substitution.) In informal terms, this example allows us to state that, if we know that a certain property P, holds for every x, and that t, stands for a particular object in our structure, then we should be able to claim P(t),. Again, we are claiming that the formulaforall x phi to phi^x_tis valid, that is, we must be able to give a "proof" of this fact, or more properly speaking, a metaproof. Actually, these examples are metatheorems of our theory of mathematical logic since we are dealing with the very concept of proof itself. Aside from this, we can also have Existential Generalization:

Axiom scheme for Existential Generalization. Given a formula phi, in a first-order language mathfrak{L},, a variable x, and a term t, that is substitutable for x, in phi,, the formula

phi^x_t to exists x phi

is universally valid.

Non-logical axioms

Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate.

Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that in principle every theory could be axiomatized in this way and formalized down to the bare language of logical formulas. This turned out to be impossible and proved to be quite a story (see below); however recently this approach has been resurrected in the form of neo-logicism.

Non-logical axioms are often simply referred to as axioms in mathematical discourse. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

Examples

This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms.

We have a language mathfrak{L}_{NT} = {0, S}, where 0, is a constant symbol and S, is a unary function and the following axioms:

forall x. lnot (Sx = 0)

forall x. forall y. (Sx = Sy to x = y)

((phi(0) land forall x.,(phi(x) to phi(Sx))) to forall x.phi(x) for any mathfrak{L}_{NT}, formula phi, with one free variable.

The standard structure is mathfrak{N} = langleN, 0, Srangle, where N, is the set of natural numbers, S, is the successor function and 0, is naturally interpreted as the number 0.

Euclidean geometry

Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry. The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist. These choices give us alternative forms of geometry in which the interior angles of a triangle add up to less than, exactly, or more than a straight line respectively and are known as elliptic, Euclidean, and hyperbolic geometries.

Real analysis

The object of study is the real numbers. The real numbers are uniquely picked out (up to isomorphism) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires use of second-order logic. The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in non-standard analysis.

Role in mathematical logic

Deductive systems and completeness

A deductive system consists, of a set Lambda, of logical axioms, a set Sigma, of non-logical axioms, and a set {(Gamma, phi)}, of rules of inference. A desirable property of a deductive system is that it be complete. A system is said to be complete if, for all formulas phi,

if Sigma models phi then Sigma vdash phi

that is, for any statement that is a logical consequence of Sigma, there actually exists a deduction of the statement from Sigma,. This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly-used type of deductive system.

Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms Sigma, of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement phi, such that neither phi, nor lnotphi, can be proved from the given set of axioms.

There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.

Further discussion

Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born. In the modern view we may take as axioms any set of formulas we like, as long as they are not known to be inconsistent.